If the first day of your last period lasting 4 days, on a 31 day cycle, the approx day the egg is coming down would be 27th of sept, with 2-4 days before that being high chance of being pregnant. Intercourse on the 30th would mean the sperm had to survive for 3 days, which is likely. Sperm can survive for about 5 days

Some conditions make your menses irregular, birth controls pills too, it's normal to not have any eggs go down once or twice a year, so I would check with your Dr, OB

You should test on the day of your missed period, technically. Blue dye and dollar test (cheapest) can give false results, pink dye are more accurate (First Response, Early Result says up to 6 days before missed period, but it is still not close to accurate) so on the day of the missed period